PHY132H1/Unit 4

Unit 4 · Introduction to Physics II

Current & Circuits

Charges in motion, and the most immediately practical unit of the course. Two conservation laws — charge at a junction, energy around a loop — solve every circuit you will meet, however complicated it looks.

By the end you can

• Relate current, voltage and resistance.
• Combine resistors in series and parallel.
• Apply Kirchhoff's junction and loop rules.
• Compute power dissipated in a circuit element.

Why it matters & where it comes from

Why it exists

Static charges are a special case. Almost everything electricity is used for involves charges flowing, and circuits are how that flow is directed and controlled.

In the real world

Household wiring is parallel, so each appliance gets the full voltage and switching one off leaves the others running. Fuses and breakers exist because power dissipated goes as $I^2R$, and too much current melts things.

Story & open questions

Ohm published his law in 1827 to widespread rejection; it was thought too simple to be true. Kirchhoff stated his rules in 1845 while still a student, and they remain the basis of all circuit analysis.

Current, resistance, Ohm's law

Definition

Current $I$ is charge per unit time, $I = Q/t$, in amperes.

Resistance $R$ measures opposition to flow, in ohms.

Ohm's law: $V = IR$ for materials where $R$ is constant.

Conventional current runs backwards

By convention current flows from positive to negative — the direction positive charges would move. In a metal wire the actual carriers are electrons, moving the opposite way. Every circuit formula uses the convention, so use it too and do not try to reconcile them.

Example Drift speed is astonishingly slow

In a typical wire carrying a few amps, individual electrons drift at well under a millimetre per second.

Yet a light turns on instantly, because the field propagates through the wire at nearly light speed and starts every electron moving at once.

The analogy is a full pipe: push water in one end and it comes out the other immediately, though no individual molecule made the trip.

Rule Power

$P = IV$ in general.

$P = I^2R$ and $P = V^2/R$ for a resistor, by substituting Ohm's law.

Use whichever form matches what you know. The $I^2R$ version is why transmission lines run at high voltage and low current: halving the current quarters the loss.

Series and parallel

Rule Series

One path, so the current is the same through each.

Voltages add: $V = V_1 + V_2$.

$R = R_1 + R_2$, always larger than either.

Rule Parallel

Branches share endpoints, so the voltage is the same across each.

Currents add: $I = I_1 + I_2$.

$\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$, always smaller than either.

These are opposite to capacitors

Resistors add in series; capacitors add in parallel. If a rule feels familiar, check which component you are dealing with. Mixing them is the single most common error in this material.

Example Why parallel lowers resistance

Adding a second path gives the current somewhere else to go, so more flows for the same voltage — and more current at the same voltage means less resistance.

Two identical resistors in parallel give exactly half. Three give a third.

A useful check: the parallel combination is always less than the smallest branch. If your answer is larger, the algebra is wrong.

Example A mixed network

A $6\,\Omega$ resistor in series with two $4\,\Omega$ resistors that are in parallel with each other, across $12$ V.

Parallel pair: $1/R = 1/4 + 1/4$, so $R = 2\,\Omega$.

Total: $6 + 2 = 8\,\Omega$, so $I = 12/8 = 1.5$ A from the battery.

Across the parallel section: $V = (1.5)(2) = 3$ V, so each branch carries $3/4 = 0.75$ A — which correctly sums back to $1.5$ A.

Kirchhoff's rules

Rule Junction rule — charge conservation

The total current into a junction equals the total current out. Charge cannot accumulate at a point.

Rule Loop rule — energy conservation

Around any closed loop, the potential changes sum to zero. Returning to where you started means returning to the same potential.

Across a battery from $-$ to $+$: gain $+\mathcal{E}$.

Across a resistor in the direction of current: drop $-IR$.

Against the current through a resistor: gain $+IR$.

Example Applying them

Two batteries and three resistors in a two-loop circuit.

Assign a current with an assumed direction to each branch. Guessing wrongly is fine — the answer comes out negative and that tells you the true direction.

Write the junction rule at each junction, giving equations relating the currents.

Write the loop rule around enough independent loops to match the number of unknowns.

Solve the resulting linear system. That is the entire method, and it works no matter how tangled the circuit.

A negative current is information, not an error

It means the real current flows opposite to your assumed direction. Do not go back and redo the problem — the magnitude is correct and the sign has told you what you wanted to know.

Common pitfalls

• Using the capacitor combination rules for resistors.
• Assuming current is "used up" as it goes round. It is the same all the way round a series loop.
• Forgetting that parallel resistance is less than the smallest branch.
• Getting the sign wrong crossing a resistor against the current.
• Treating a negative current as a mistake.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Ohm's law

Find the current through a $220\,\Omega$ resistor at $12$ V.

Answer

$I = 12/220 = 0.055$ A, or $55$ mA.

Practice 2 · Series

Three $10\,\Omega$ resistors in series. Find the total.

Answer

$30\,\Omega$.

Practice 3 · Parallel

Three $10\,\Omega$ resistors in parallel. Find the total.

Answer

$1/R = 3/10$, so $R = 3.33\,\Omega$ — less than the smallest branch, as required.

Practice 4 · Power

How much power does a $60\,\Omega$ heater draw at $120$ V?

Answer

$P = V^2/R = 14400/60 = 240$ W.

Practice 5 · Household wiring

Why are home outlets wired in parallel?

Answer

So every appliance gets the full supply voltage and switching one off does not break the circuit for the rest. In series they would share the voltage and all fail together.

Practice 6 · Transmission

Why is power transmitted at high voltage?

Solution

For a given power $P = IV$, raising $V$ lowers $I$. Line losses go as $I^2R$, so halving the current cuts the loss to a quarter. The voltage is stepped back down by transformers near the point of use.

Practice 7 · Loop rule

A $9$ V battery drives current through $2\,\Omega$ and $4\,\Omega$ in series. Verify the loop rule.

Solution

$I = 9/6 = 1.5$ A. Drops are $(1.5)(2) = 3$ V and $(1.5)(4) = 6$ V. Going round: $+9 - 3 - 6 = 0$. ✓

Practice 8 · Negative answer

You solve a circuit and get $I = -0.4$ A. What does that mean?

Answer

The current is $0.4$ A flowing opposite to the direction you assumed. The magnitude is right and no rework is needed.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY132H1 topic map. It should cover the following items before you move on:

• Current and resistance. Definitions, Ohm's law, and the conventional-current convention.
• Combinations. Series and parallel, and how they differ from capacitors.
• Kirchhoff's rules. Junction and loop, with correct signs.
• Power. The three forms and when each is convenient.

Textbook/source alignment

Circuits fall under the calendar's "electricity" topic and dominate the laboratory component. Check the syllabus on Quercus for the assigned problems and for which lab experiments accompany this material.

Core idea map

Potential differences from Unit 3 drive current; resistance opposes it; charge and energy conservation give two rules that solve any network. Unit 5 asks what those moving charges do to the space around them.

Study routine. Redraw every circuit before solving, grouping obvious series and parallel blocks. Most circuit problems that look hard are simple ones drawn awkwardly.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · Series current

Does current decrease as it passes through resistors in series?

Guide

No. Charge is conserved, so the same current flows throughout. The voltage drops, not the current.

Extension 2 · Parallel check

What is a quick sanity check on a parallel combination?

Guide

It must be smaller than the smallest branch resistance.

Extension 3 · The two rules

What conservation law does each Kirchhoff rule express?

Guide

Junction: conservation of charge. Loop: conservation of energy.

Extension 4 · Drift speed

Why does a lamp light instantly if electrons drift slowly?

Guide

The electric field establishes itself through the wire at close to light speed, so every electron in the circuit starts moving essentially at once.

← Unit 03Electric potential Unit 05 →Magnetic fields